College Algebra: Radical Expressions w Variables

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Okay, so we fooled around with looking at radicals and all these fancy exponents of fractions and stuff with numbers. But, in fact, the exact same thing also works if you have variables there's really no difference. So let's just try a couple of examples just for me to convince you that really there is no difference at all, it's the exact same method.
So let's take a look at the following, how about look at the square root, a big square root, , and let's see how far we can simplify that and make that sort of tidy looking. Well, the first thing I remember is that I can rewrite this as because by a power that is the same thing as taking a square root. Okay, well, now I remember that if I have a whole bunch of different things all raised to the same power one of the laws that we saw of exponents is that I can just do it sort of ala carte, take this to that power, multiply it by this to that power, take it by that power, one of the laws we saw. So in fact, this is the same thing as 25(x^6)(y^5). And now I can sort of think about these things individually. Notice that is always the same thing, divide and conquer as much as possible. Now, you might want to rewrite this, because 25 is actually the same thing as or hopefully, at some point, you will get so comfortable with that you will just say, oh, it's the same thing and so I know is 5.
Now here we can actually do a little simplification, because remember a different rule of exponents which says that if you have something to a power and you raise the whole thing to another power, what do you do with the exponents? If you think about it, you multiply them, right. If you are not sure just go back and take like A^3 and square it and see what happens and then you will see you'll have A^6. So here I take 6 and multiply it by a so take half of 6, which is going to be x and then I've got y. And I could simplify that even further and say this is just 5x^3 and now I could keep this as y and now basically this all, you are on your own. I don't know how you want to say it.
I'll show you one funky way of saying this, which maybe you will like or maybe you won't, I don't know. If it were me, by the way, I would be happy with y that's fine. If it is not me and it might be someone else you could write it this way, we need to do this in two steps y^2^^^^^ and then using laws of exponents I could remember that if I'm adding exponents that means that I'm multiplying bottoms. So, in fact, this just equals 5x^3 and then here I'm going to see a y^2 times y^, which is the . So it looks pretty different, but really it is the same thing, because this is the y^^, this is the y^2 and I'm combining them by adding exponents. There you go.
Okay, let's try another one. Now, if you think that was bad wait until you see this one let's see if you can keep your seats. Let's take the , but we're not done yet, we haven't even started yet. Then we take think that's it? Not even close, because I'm going to divide all of that by the fourth root . Now that is a problem. That is a biggee. And I want to see if we can simplify that down to something that is a little bit more manageable. Okay, now what's my thinking process here? My thinking process is actually pretty straightforward. I see a lot of fourth roots everywhere, so I'm going to use properties of exponents when I start to think about this is this to the ^th power and this to the ^th power and so forth. To combine this as to one big fourth power, see, in fact this is just one big fourth power I could get rid of that, look at that, I can get rid of that and put a line there and it's perfect. You see that? Okay, and, then I can say well a fourth power on the top and a fourth power on the bottom that's just one humongoid fourth power, I mean, just a huge 4 with a one big thing and get rid of all of them.
So what I'm doing is I'm trying to get rid of all that stuff and write it as just that complicated fraction all to the one-fourth power and that's what I'm going to do. So let's write this as a complicated fraction so that's . See, I pulled off all those fourth roots into one big thing. Okay, now what? Now I can just cancel away if I'm really careful, because I can cancel any of the common factors on top and bottom. For example, here I have an r^3 and here I see r^6, remember what you do, you subtract the exponents or if you don't like that just write it out. There are six r's here, three of them here, I cancel them and I put three more on top, so if I cancel this away with this I'm left with an r^3.
I've got some messes on top and some messes on the bottom. And they are both factors of the top and bottom I can cancel. In fact I can cancel all of these with half of those and I will just be left with instead of s^4 I will be left now with s^2. And, I've got some factors of 2 around here I can cancel. This 4 I can cancel with that 2, but I still have a 2 downstairs and I can take that 2 and cancel it with the 32 a little bit and that would give me something like 16, yeah, 16 x 2 = 32.
So now it looks like there is a lot of mess here so it's good to always be very careful. By the way, when you are doing these kind of really hideous problems that you wonder why anyone would ever assign, just remember to write very large. I know that sounds stupid, but the larger you write, if you start writing really tiny then twos becomes threes and threes become nines, just write large and write neatly and you will be amazed at the accuracy you'll achieve.
Okay, here we go. So I see 16 and let's see, I've got an r^3 here, I also have an r here, if I combine them using laws of exponents I see r to the fourth. And then here I just see an s and there is an s^2 so I see an s^3, that looks like the top to me. And what's the bottom? Well, the bottom, there is nothing there, so does that mean zero? No. Remember there's always an invisible 1 down there, an invisible 1 factor. So, in fact, I just put that all over 1, or I could actually just ignore writing anything, because anything over 1 is just itself, but I'll put it in there just for laughs. And now what happens? Well, now I have to take the fourth root of all that. So, you can do that ala carte again, and I don't know, do you want me to write it like fourth root, or do you want me to write it as a one-fourth exponent I don't know? I wish you'd tell me. I'll write it like this just for fun. I'll use the exponents again. Each term gets a fourth root to it, every single term, we're not going to leave anyone out. Okay, now what happens here? Well, 16^ so what does that mean? I have to think of the fourth root of 16. That is some number that has the property that it times itself times itself times itself is going to equal 16. Well, 2 works, so this equals 2. Now, what is r^, well, you can either think about that or just realize that you can cancel these out into just r. And this, well, sadly there's not much I can tell you about that. I'm going to keep that as just s^, so that is the answer. You could say it that way or if you want to say it in a radical way, you could say the fourth root of s and then cube that number. Either answer is great for me. Quite frankly, I like this one better.
Anyway, either one is a dramatic simplification from this original horrific thing. Now we've got this. So, you can see that factoring and using the exponents and radicals and so forth with variables it's not a big deal. All we have to do is use very careful bookkeeping and cancel very carefully all the common factors and we're home free.
Prerequisites
Radical Expressions
Simplifying Radical Expressions with Variables Page [1 of 2]